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<div id="Sx1" class="ltx_section">
<h1 class="ltx_title ltx_title_section">Threshold selection</h1>

<div id="Sx1.p1" class="ltx_para">
<p class="ltx_p">When finding the approximate position of molecules, choosing the right
threshold value is important for obtaining good results, because the
threshold influences the number of missed molecules (false negatives)
and the number of erroneous detections of non-existing molecules (false
positive detections). ThunderSTORM uses a single-valued intensity
threshold which is updated for every raw input image <img id="Sx1.p1.m1" class="ltx_Math" style="vertical-align:-2px" src="mi/mi16.png" width="13" height="16" alt="I"> and applied
to the filtered image <img id="Sx1.p1.m2" class="ltx_Math" style="vertical-align:-2px" src="mi/mi2.png" width="18" height="16" alt="F">. The threshold value can be specified by
users as an expression combining mathematical functions and operators
with variables based on the current raw or filtered image. This is
a powerful option, because users can specify the threshold value systematically
for unknown input images, in which the global intensity may slowly
fluctuate over time.</p>
</div>
<div id="Sx1.SSx1" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">Guidelines for the choice of parameters</h2>

<div id="Sx1.SSx1.p1" class="ltx_para">
<p class="ltx_p">Our experiments indicate that the <a href="LocalMaximaDetector.html" title="" class="ltx_ref">local maximum</a>
approach with 8-connected neighborhoods provides the highest <a href="../PerformanceEvaluationPlugIn.html" title="" class="ltx_ref">F1-score</a>
compared to <a href="Detectors.html" title="" class="ltx_ref">other methods</a>. The suggested threshold
applied to the filtered images, in the case of the <a href="../filters/ui/CompoundWaveletFilterUI.html" title="" class="ltx_ref">wavelet filter</a>,
usually ranges from 0.5 to 2 times the standard deviation of the 1st
wavelet level, e.g., <span class="ltx_text ltx_font_typewriter">1*std(Wave.F1)</span>. Such a value is recommended
by Izeddin et al. <cite class="ltx_cite">[<a href="#bib.bib16" title="Wavelet analysis for single molecule localization microscopy" class="ltx_ref">1</a>]</cite> and works well for typical SMLM
data. Increasing the threshold value will lead to less false positive
detections at the expense of more missed molecules and vice versa.</p>
</div>
</div>
<div id="Sx1.SSx2" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">Formula interpreter syntax</h2>

<div id="Sx1.SSx2.p1" class="ltx_para">
<p class="ltx_p">The formula interpreter provides several built-in statistical functions
and some predefined variables. A brief description of the syntax and
semantic rules follows.</p>
</div>
<div id="Sx1.SSx2.SSSx1" class="ltx_subsubsection">
<h3 class="ltx_title ltx_title_subsubsection">Variables</h3>

<div id="Sx1.SSx2.SSSx1.p1" class="ltx_para">
<p class="ltx_p">Variables used in formulas can be scalars (numbers) or images (matrices).
Image variables are provided by image filters. Two variables are always
present:</p>
</div>
<div id="Sx1.SSx2.SSSx1.p2" class="ltx_para">
<table class="ltx_tabular ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left"><span class="ltx_text ltx_font_typewriter">I</span></td>
<td class="ltx_td ltx_align_left">current raw input image</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left"><span class="ltx_text ltx_font_typewriter">F</span></td>
<td class="ltx_td ltx_align_left">corresponding filtered image</td>
</tr>
</tbody>
</table>
</div>
<div id="Sx1.SSx2.SSSx1.p3" class="ltx_para">
<p class="ltx_p">Note that some of the filters can provide more variables, see help
of individual <a href="../filters/ui/Filters.html" title="" class="ltx_ref">image filters</a> for
possible variable names. The formula interpreter can also use any
of the variables provided by image filters even if the filter was
not selected for image filtering.</p>
</div>
<div id="Sx1.SSx2.SSSx1.p4" class="ltx_para">
<p class="ltx_p">For example, suppose that a Gaussian filter is selected as an image
filter and the threshold for the approximate localization of molecules
is defined by the formula<span class="ltx_text ltx_font_typewriter"> mean(Med.F)+10*std(Wave.F1)</span>.
The image filtering operation will run several times in this case,
because the formula interpreter will determine the mean intensity
value of the input image filtered using a median filter and will add
ten times the standard deviation of intensity values obtained as a
first wavelet level of the input image. This threshold value will
be applied to the input image filtered using the selected Gaussian
filter.</p>
</div>
</div>
<div id="Sx1.SSx2.SSSx2" class="ltx_subsubsection">
<h3 class="ltx_title ltx_title_subsubsection">Operators</h3>

<div id="Sx1.SSx2.SSSx2.p1" class="ltx_para">
<p class="ltx_p">Operators can take images or scalars as operands. All image operations
are performed element-wise, therefore image matrices must have the
same size. The semantics is as follows:</p>
</div>
<div id="Sx1.SSx2.SSSx2.p2" class="ltx_para">
<table class="ltx_tabular ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left"><span class="ltx_text ltx_font_typewriter">scalar + scalar = scalar</span></td>
<td class="ltx_td"></td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left"><span class="ltx_text ltx_font_typewriter">image + image = image</span></td>
<td class="ltx_td"></td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left"><span class="ltx_text ltx_font_typewriter">image + scalar = image</span></td>
<td class="ltx_td ltx_align_left">(scalar is added to each element of the image)</td>
</tr>
</tbody>
</table>
</div>
<div id="Sx1.SSx2.SSSx2.p3" class="ltx_para">
<p class="ltx_p">The supported operators are:</p>
</div>
<div id="Sx1.SSx2.SSSx2.p4" class="ltx_para">
<table class="ltx_tabular ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left"><span class="ltx_text ltx_font_typewriter">a + b</span></th>
<td class="ltx_td ltx_align_left">addition</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left"><span class="ltx_text ltx_font_typewriter">a - b</span></th>
<td class="ltx_td ltx_align_left">subtraction</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left"><span class="ltx_text ltx_font_typewriter">a * b</span></th>
<td class="ltx_td ltx_align_left">multiplication</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left"><span class="ltx_text ltx_font_typewriter">a / b</span></th>
<td class="ltx_td ltx_align_left">division</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left"><span class="ltx_text ltx_font_typewriter">a % b</span></th>
<td class="ltx_td ltx_align_left">modulo</td>
</tr>
<tr class="ltx_tr">
<th class="ltx_td ltx_align_left"><span class="ltx_text ltx_font_typewriter">a ^ b</span></th>
<td class="ltx_td ltx_align_left">exponentiation (<span class="ltx_text ltx_font_typewriter">b</span> must be a scalar)</td>
</tr>
</tbody>
</table>
</div>
</div>
<div id="Sx1.SSx2.SSSx3" class="ltx_subsubsection">
<h3 class="ltx_title ltx_title_subsubsection">Functions</h3>

<div id="Sx1.SSx2.SSSx3.p1" class="ltx_para">
<p class="ltx_p">Functions can take an image or a scalar as an argument or a result
of a function which is an image or a scalar. The supported functions
are:</p>
</div>
<div id="Sx1.SSx2.SSSx3.p2" class="ltx_para">
<table class="ltx_tabular ltx_align_middle">
<tbody class="ltx_tbody">
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left"><span class="ltx_text ltx_font_typewriter">var(x)</span></td>
<td class="ltx_td ltx_align_left">variance of <span class="ltx_text ltx_font_typewriter">x</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left"><span class="ltx_text ltx_font_typewriter">std(x)</span></td>
<td class="ltx_td ltx_align_left">standard deviation of <span class="ltx_text ltx_font_typewriter">x</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left"><span class="ltx_text ltx_font_typewriter">mean(x)</span></td>
<td class="ltx_td ltx_align_left">mean value of <span class="ltx_text ltx_font_typewriter">x</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left"><span class="ltx_text ltx_font_typewriter">median(x)</span></td>
<td class="ltx_td ltx_align_left">median of <span class="ltx_text ltx_font_typewriter">x</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left"><span class="ltx_text ltx_font_typewriter">min(x)</span></td>
<td class="ltx_td ltx_align_left">minimum value in <span class="ltx_text ltx_font_typewriter">x</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left"><span class="ltx_text ltx_font_typewriter">max(x)</span></td>
<td class="ltx_td ltx_align_left">maximum value in <span class="ltx_text ltx_font_typewriter">x</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left"><span class="ltx_text ltx_font_typewriter">sum(x)</span></td>
<td class="ltx_td ltx_align_left">sum of all items in <span class="ltx_text ltx_font_typewriter">x</span>
</td>
</tr>
<tr class="ltx_tr">
<td class="ltx_td ltx_align_left"><span class="ltx_text ltx_font_typewriter">abs(x)</span></td>
<td class="ltx_td ltx_align_left">absolute value of <span class="ltx_text ltx_font_typewriter">x</span>
</td>
</tr>
</tbody>
</table>
</div>
</div>
</div>
<div id="Sx1.SSx3" class="ltx_subsection">
<h2 class="ltx_title ltx_title_subsection">See also</h2>

<div id="Sx1.SSx3.p1" class="ltx_para">
<ul id="I1" class="ltx_itemize">
<li id="I1.i1" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">•</span> 
<div id="I1.i1.p1" class="ltx_para">
<p class="ltx_p"><a href="../filters/ui/Filters.html" title="" class="ltx_ref">Image filtering and feature enhancement</a></p>
</div>
</li>
<li id="I1.i2" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_itemize">•</span> 
<div id="I1.i2.p1" class="ltx_para">
<p class="ltx_p"><a href="Detectors.html" title="" class="ltx_ref">Finding approximate positions of molecules</a></p>
</div>
</li>
</ul>
</div>
</div>
</div>
<div id="bib" class="ltx_bibliography">
<h1 class="ltx_title ltx_title_bibliography">References</h1>

<ul id="L1" class="ltx_biblist">
<li id="bib.bib16" class="ltx_bibitem ltx_bib_article">
<span class="ltx_bibtag ltx_bib_key ltx_role_refnum">[1]</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_author">I. Izeddin, J. Boulanger, V. Racine, C. G. Specht, A. Kechkar, D. Nair, A. Triller, D. Choquet, M. Dahan and J. B. Sibarita</span><span class="ltx_text ltx_bib_year">(2012)</span>
</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_title">Wavelet analysis for single molecule localization microscopy</span>,
</span>
<span class="ltx_bibblock"><span class="ltx_text ltx_bib_journal">Optics Express</span> <span class="ltx_text ltx_bib_volume">20</span> (<span class="ltx_text ltx_bib_number">3</span>), <span class="ltx_text ltx_bib_pages"> pp. 2081–95</span>.
</span>
<span class="ltx_bibblock">External Links: <span class="ltx_text ltx_bib_links"><a href="http://dx.doi.org/10.1364/OE.20.002081" title="" class="ltx_ref doi ltx_bib_external">Document</a></span>.
</span>
<span class="ltx_bibblock ltx_bib_cited">Cited by: <a href="#Sx1.SSx1.p1" title="Guidelines for the choice of parameters ‣ Threshold selection" class="ltx_ref"><span class="ltx_text ltx_ref_title">Guidelines for the choice of parameters</span></a>.
</span>
</li>
</ul>
</div>
</div>
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